Drift conditions on a HJM model with stochastic basis spreads. Teresa Martínez Quantitative Product Group. Santander Quantitative Product Group

Transcription

1 Drif condiions on a HJM model wih sochasic basis spreads eresa Marínez Quaniaive Produc Group Sanander Quaniaive Produc Group

2 Conens 1 Inroducion 1 2 Seing of he problem. Noaion Bonds and curves he FX spo process Drifs of he curves Domesic currency discoun curve i.f.r. drif: he HJM condiion Domesic currency esimaion curve In erms of spread curve i.f.r. drif Foreign currency discoun curve i.f.r. drif Foreign currency esimaion curve An example: dynamics in a Hull-Whie model Domesic discouning bonds dynamics Domesic Libor dynamics Foreign discouning bond dynamics Foreign Libor dynamics

3 Preliminary version Sepember 17, 29 Absrac. A general HJM model is considered o model he evoluion of discoun and reference curves (in several currencies) under he risk-neural probabiliy associaed o he discoun curve of he domesic economy. Drif condiions are derived for hese dynamics ha make he model consisen wih he marke pracice of curve consrucion. Some numerical ess are included in he paricular case of he Hull-Whie model. 1 Inroducion he global economy has been suffering, since subprime morgage crisis arose in summer 27, imporan modificaions in markes all over he world. No only he volume of operaions, affeced by liquidiy and credi consrains, has experienced his impac. he whole undersanding of he rading machinery has gone under review as a byproduc of such changing siuaion. For fixed income securiies, one of he consequences of he credi crunch, among ohers, is he widening of spreads beween formerly equivalen raes (or wih negligible differences for curve consrucion purposes) for insrumens of he same mauriy bu differen underlying enors. his divergence originaes in he changes of credi and liquidiy disposal among he marke agens. A heurisical explanaion of his effec can be found in [M]. o our knowledge, here is no general heory already acceped o rigorously explain his effecs from he heoreical poin of view. Neverheless, he marke agens have already developed a way o rea his divergence, becoming a usual marke pracice he valuaion of fixed income derivaives by using differen curves: a unique curve o discoun flows, and differen ones o esimae floaing flows. In he classical one curve seing, he valuaion of he floaing paymen of a sandard marke FRA is given by a replicaion sraegy consisen in selling one bond and buying anoher one. In his seing, he floaing paymen can be expressed as [( ) B(, ) (1.1) F RA floaing (,, U) = B(, ) B(, U) = m(, U)B(, U) B(, U) 1 1 m(, U) where B(, ) sands for he( price oday) of a bond paying one uni of currency a ime (he B(, ) discoun facor a ime ), B(,U) 1 1 m(,u) is he expression of he Libor rae prevailing oday for he accrual period (, U) and m(, U) is he day coun fracion from o U according o marke convenion. his expression is derived, from he definiion of a FRA, hrough he use of well known no arbirage argumens. hese argumens replicae marke behavior as long as he marke admis his replicaion sraegy. Acually, his is no he case, and he forward raes implied from he deposi raes (equivalen o he expression appearing in (1.1)) are no ], wo curves valuaion Quaniaive Produc Group, Sanander 1

4 Preliminary version Sepember 17, 29 necessarily equal o he quoed FRA raes. hese are hen replicaed by using a differen curve, in such a way ha he quoed FRA rae corresponds o ( ) Br (, ) B r (, U) 1 1 m(, U), where B r (, ) is he discoun facor calculaed according o a special curve, buil for raes of enor U (ypically U = 1M, 3M, 6M and 12M ). hen, he price of a FRA is, in his wo curve seing, [( ) Br (, ) (1.2) F RA floaing (,, U) = m(, U)B d (, U) B r (, U) 1 1 m(, U) Any model ha inends o fi his siuaion needs o cope wih he fac ha he usual no arbirage argumens have o be carefully applied. In he presen work, we consider he problem of defining a HJM model consisen wih his marke pracice. More concreely, we assume ha under he discoun bank accoun risk-free probabiliy, P d wih associaed numeraire e r d(s) ds, he insananeous forward raes (i.f.r. from now on) of he discoun and reference curves, f d (, ) and f r (, ), respecively, follow he dynamics df d (, ) = µ d (, )d + σ d (, )dw d, r d () = f d (, ) df r (, ) = µ r (, )d + σ r (, )dw r, r r () = f r (, ). By classical no-arbirage heory, i is known ha (1.3) µ d (, ) = σ d (, ) σ d (, s)ds. Our aim is finding he expression for he drif of he reference curve, in such a way ha ( (1.4) F RA floaing (,, U) = E P e ) U r d (s) ds L r (,, U) = B d (, U)L r (,, U), where L r (,, U) is he Libor fixing a wih accrual period (, U), whose expression we assume o be L r (,, U) = ( ) Br (, ) B r (, U) 1 1 ( e m(, U) = U ) fr(, ) d 1 ], 1 m(, U). Observe ha his requiremen is a naural exension of he marke definiion of reference curves, ha ranslaes ino expressions like (1.2). Of course, (1.4) is saed for each reference curve, only for floaing paymens of he Libor index associaed o he curve. Also, we consider his problem in he muli currency seing, giving he expressions of he drif of he foreign discoun and reference curves under he domesic discouning risk-neural probabiliy. I is convenien o noe ha we are no dealing wih he consrucion of reference curves, somehing ha consiues a research opic in iself. We assume he exisence of a well defined se of curves, wih appropriae inerpolaion and boosrapping algorihms, and only give condiions on he diffusions of he i.f.r. s dynamics o mee condiion (1.4). wo curves valuaion Quaniaive Produc Group, Sanander 2

5 Preliminary version Sepember 17, 29 As far as we know, his approach o he wo curves valuaion in he HJM seing is new. here are precedens of HJM models exensions o valuae credi spreads, by posulaing he evoluion of he defaulable bonds as given by a defaulable i.f.r.wih an HJM dynamics. See, for example, [DS], [S2] and [ChFM] (see also [S1] for a similar approach for a Libor Marke Model). In all hese references, he modeling of he (defaulable) raes is ruled by credi consrains. Al hough he origin of he exisence of reference curves is he already menioned credi and liquidiy crisis, i is no clear how o incorporae i in he diffusion of reference raes by using a similar approach. In [JP] and [AS], general frameworks o incorporae liquidiy risk in he valuaion of marke securiies are sudied. hey exend classical no arbirage pricing heory o an economy in which he securiies price depends also on he size of he rade considered. Basis spreads among raes for insrumens of he same mauriy denominaed in he same currency have been considered negligible unil summer 27 credi crisis. Before his poin, basis spreads for raes have been mainly sudied in he muli currency case, o price cross currency derivaives, see for insance [BS] and []. he appearance of basis spreads among one currency marke raes is reaed in [Bi] by using an FX analogy. his mehodology is used o derive prices of vanilla producs (cap/floor, swaps and swapions) in a one currency muli curve seing, and produces valuaion formulae wih quano like adjusmens. Unadjused formulae are derived in [M] in an exension of Libor Marke Models o he muli curve seing. In [KW], a hree curve model (wih one-facor quadraic Gaussian dynamics) is proposed, o sudy opion prices on swaps and bonds. he organizaion of he paper is as follows. he noaion o be used hroughou he paper is saed in secion 2, in secion 3 he main resuls are saed, and in secion 4, an applicaion o a Hull-Whie model is given. 2 Seing of he problem. Noaion Our aim is being able o derive a muli curve muli currency HJM model in a consisen way wih he marke pracice. o his aim, we consider a domesic economy ruled by a discoun curve and several reference curves, such ha floaing paymens depend on raes relaed o he reference curves, and are discouned wih he discoun reference curve. he i.f.r. for he reference curve can be expressed as he sum of he discouning i.f.r. plus a spread i.f.r., ha we will also consider in he model. In he foreign economy, a similar siuaion happens, and we have a FX spo process o change from one foreign economy o he domesic one. For simpliciy, we will consider only a reference curve in he domesic economy, only one foreign economy, and only one reference curve in he foreign economy (all he resuls would apply in he general case, simply repeaing he argumens wih each domesic reference curve and each foreign economy). In his siuaion, we will consider he following noaion for he differen i.f.r.: wo curves valuaion Quaniaive Produc Group, Sanander 3

6 Preliminary version Sepember 17, 29 fd d(, ) f f d (, ) fr d (, ) fr f (, ) fs d (, ) fs f (, ) discouning i.f.r.in he domesic economy discouning i.f.r. in he foreign economy reference i.f.r. in he domesic economy reference i.f.r. in he foreign economy spread i.f.r. in he domesic economy f d r (, ) = f d d (, ) + f d s (, ) spread i.f.r. in he foreign economy f f r (, ) = f f d (, ) + f f s (, ) All of hese raes live and evolve in a filered space (Ω, F, P d, {F }), rich enough o carry a P d Brownian moion of he suiable dimension, where P d is he domesic risk neural probabiliy. We will consider ha each of hese raes saisfy an Iô diffusion, df α γ (, ) = µ α γ (, ) d + σ α γ (, )dw αγ wih α = d, f (domesic, foreign) denoing he economy we are referring o, γ = d, r, s (discoun, reference, spread) denoing he curve in each economy and W αγ a Brownian moion under he domesic risk neural probabiliy. Also, we wrie ρ αγ α γ () d = d W αγ, W α γ he insananeous correlaion beween any wo of he Brownian moions driving i.f.r. s diffusions. We assume ha all he processes depending on a mauriy parameer are smooh enough o be differeniable and inegrable w.r. his parameer, and Fubini s heorem holds. 2.1 Bonds and curves We will ofen use ha under each curve, he bond price process is given by (2.1) B α γ (, ) = e f α γ (,s) ds, α = d, f, γ = d, r, s. Le us forge for a momen he super and subscrips α, γ, since he following calculaion is valid in any case, and define Y (, ) = f(, s)ds, such ha he bond price can also be compued as B(, ) = e Y (, ). For his process, we have Y (, ) = = [f(, s) + f(, s) ds + µ(u, s) du + µ(u, s) du ds µ(u, s) du ds + σ(u, s) dw u ] ds σ(u, s) dw u ds. σ(u, s) dw u ds wo curves valuaion Quaniaive Produc Group, Sanander 4

7 Preliminary version Sepember 17, 29 Noe ha µ(u, s) du ds = = u µ(u, s) ds du µ(u, s) ds du + u µ(u, s) ds du, and he same happens for he oher inegrals. By subsiuing hese expressions in he former equaion, and simplifying he repeaed erms, we ge Y (, ) = f(, s) ds + f(, s) ds u u By changing he order of inegraion, we ge µ(u, s) ds du + µ(u, s) ds du u u σ(u, s) ds dw u σ(u, s) ds dw u. By observing ha and defining: u u µ(u, s) ds du = σ(u, s) ds dw u = f(, ) = f(, ) + s s µ(s, ) ds + µ(u, s) du ds σ(u, s) dw u ds. σ(s, ) dw s (2.2) A(, ) = we ge Y (, ) = Finally, we have Y (, ) B(, ) = e { (2.3) = exp And by Iô s rule, f(, s) ds + µ(, s)ds, S(, ) = f(, s) ds A(u, )du + A(u, )du σ(, s)ds, S(u, )dw u S(u, )dw u + f(s, s)ds. } f(s, s)ds.. (2.4) db(, ) B(, ) = ( f(, ) A(, ) (S(, ))2) d S(, ) dw Anoher useful expression is he dynamics of Φ = B 1 (, S) B 2(, ) B 2 (, U) wo curves valuaion Quaniaive Produc Group, Sanander 5

8 Preliminary version Sepember 17, 29 wih B 1 and B 2 defined as in (2.1) for wo differen i.f.r., f 1 and f 2. In his siuaion, we can wrie where φ = A 1 (u, S)du + Φ = Φ e φ S 1 (u, S)dW 1 u + (A 2 (u, U) A 2 (u, )) du + f 1 (s, s)ds + (S 2 (u, U) S 2 (u, ))dw 2 u. Anoher applicaion of Iô s rule gives dφ = dφ + 1 Φ 2 d φ ( = f 1 (, ) A 1 (, S) + A 2 (, U) A 2 (, ) ) (2.5) (S 1(, S)) (S 2(, U) S 2 (, )) 2 ρ 1 2()S 1 (, S)(S 2 (, U) S 2 (, )) d S 1 (, S)dW 1 + (S 2 (, U) S 2 (, ))dw he FX spo process he FX spo processes S is defined as he number of unis of domesic currency per one uni of foreign currency, saisfying ds (2.6) = (fd d S (, ) f f d (, )) d + σs () dw S, wih ρ αγ S () = d W αγ, W S he insananeous correlaion beween he Brownian moions of he i.f.r. and FX spo. In our calculaions, only ρ fd S will appear. 3 Drifs of he curves 3.1 Domesic currency discoun curve i.f.r. drif: he HJM condiion hroughou his secion we consider fd d (, ) := f(, ) = f(, ) + µ(s, ) ds + By (2.4), in order o avoid arbirage we mus impose: (3.1) A(, ) = 1 2 S2 (, ). σ(s, )dw s. Differeniaing boh sides wih respec o, he former condiion is equivalen o require he domesic discouning i.f.r. drif saisfies: (3.2) µ(, ) = σ(, ) σ(, s)ds. wo curves valuaion Quaniaive Produc Group, Sanander 6

9 Preliminary version Sepember 17, Domesic currency esimaion curve We consider a model for domesic floaing paymens ruled by wo curves, a discoun curve and a reference curve, whose i.f.r. are given, respecively, by: df d d (, ) := df d (, ) = µ d (, ) d + σ d (, ) dw d, df r d (, ) := df r (, ) = µ r (, ) d + σ r (, ) dw r, wih d W d, W r = ρ d r() d. he floaing paymens considered are such ha he discoun of he flow is calculaed wih he discoun curve, and he applicable Libor rae is calculaed wih he reference curve. We will also consider he formulaion of he problem in erms of he spread curve, whose i.f.r. is buil as he difference beween he discoun and he reference i.f.r. s, namely We denoe by L r (,, U) = f s (, ) = f d (, ) f r (, ). ( Br ( ), ) B r (, U) 1 1 ( e m (, U) = U ) fr(, ) d 1 he Libor rae fixing a, wih naural accrual period (, U). 1 m (, U) Our aim is finding a suiable expression for he drif of he reference (or spread) i.f.r. such ha he expeced value a any ime of he discouned floaing flow is given by (1.4). Since we can wrie ( E P e ) U f d (s,s) ds L r (,, U) = E P condiion (1.4) is saisfied if he process ( e ) f d (s,s) ds B d (, U)L r (,, U), X u = e u f d(s,s) ds B d (u, U)L r (u,, U) is maringale under he risk neural measure. his will be our assumpion. Noe ha his condiion is immediaely saisfied, by he usual no arbirage heory, if B d (u, U)L r (u,, U) represens he price of a radable asse 1. Bu we are no posulaing his, insead, we are only imposing a condiion on he dynamics of he process. In fac, i is enough if we require he process (3.3) Z = B d (, U) B r(, ) B r (, U) = B d(, ) B s(, ) B s (, U) o saisfy an Iô diffusion of he form dz Z = f d (, ) d + (a maringale erm). 1 A ime u, L r(u,, U) is known, and he quaniy B d (u, U)L r(u,, U) is a fixed number of conracs paying a uni of currency a ime U. wo curves valuaion Quaniaive Produc Group, Sanander 7

10 Preliminary version Sepember 17, 29 By (2.5) wih f 1 = f d, f 2 = f r, S = U, we have dz = f d (, ) d + ( A r (, U) A r (, ) A d (, U) ) d Z + 1 [ ] (3.4) (S r (, U) S r (, )) 2 + (S d (, U)) 2 2ρ r 2 s()s d (, U)(S r (, U) S r (, )) d +(S s (, U) S s (, )) dw s S d (, U)dW d. Condiion (1.4), by using (3.2), means ha we are imposing U (3.5) µ r (, ) d = 1 2 (S r(, U) S r (, )) 2 + ρ d r()s d (, U)(S r (, U) S r (, )) U U = σ r (, ) σ r (, ) d d + ρ d r() σ d (, ) d σ r (, ) d. We wrie U = + δ (δ he enor of he Libor index associaed o he reference curve), and assume (3.5) holds for any wih +δ µ r (, ) d = g r (, ), g r (, ) = +δ for any. In paricular, choosing =, +δ +δ σ r (, ) σ r (, ) d d + ρ d r() σ d (, ) d σ r (, ) d, +δ +δ +δ g r (, ) = σ r (, ) σ r (, ) d d + ρ d r() σ d (, ) d σ r (, ) d, and here are an infiniy of possible choices for he value of µ r (, ) wih [, + δ) saisfying for example +δ µ r (, ) d = g r (, ), µ r (, ) = 1 δ g r(, ). or +δ µ r (, ) = σ r (, ) σ r (, ) d + ρ d r()σ r (, ) σ d (, ) d. or +δ µ r (, ) = σ r (, ) σ r (, ) d + ρ d r()σ d (, ) σ r (, ) d. For + δ, we can differeniae in our condiion o ge ha µ r should saisfy µ r (, + δ) µ r (, ) = g r (, ), wo curves valuaion Quaniaive Produc Group, Sanander 8

11 Preliminary version Sepember 17, 29 wih g r (, ) = (σ r(, + δ) σ r (, )) +ρ d r()σ d (, + δ) +δ For [ + iδ, + (i + 1)δ), we can wrie +δ σ r (, ) d d σ r (, ) d + ρ d r() +δ σ d (, ) d (σ r (, + δ) σ r (, )). µ r (, ) = µ r (, δ) + g r (, δ) = µ r(, 2δ) + g r (, δ) + g r (, 2δ) =... = i g r µ r (, iδ) + (, jδ). j=1 Wih his expression, we easily have ha condiion (3.5) is saisfied for any choice of he expression of µ r (, ) when is in [, + δ) In erms of spread curve i.f.r. drif By (2.5) wih f 1 = f d, f 2 = f s, S =, we have ha he process (3.3) saisfies dz = f d (, ) d + ( A s (, U) A s (, ) A d (, ) ) d Z + 1 [ ] (3.6) (S s (, U) S s (, )) 2 + (S d (, )) 2 2ρ d 2 s()s d (, )(S s (, U) S s (, )) d +(S s (, U) S s (, )) dw s S d (, )dw d. Condiion (1.4), by using (3.2), means ha we are imposing U (3.7) µ s (, ) d = 1 2 (S s(, U) S s (, )) 2 + ρ d s()s d (, )(S s (, U) S s (, )) U U = σ s (, ) σ s (, ) d d + ρ d s() σ d (, ) d σ s (, ) d. We wrie U = + δ wih δ he enor of he index considered a fixed amoun of ime, and assume (3.7) wih +δ µ s (, ) d = g s (, ), +δ +δ g s (, ) = σ s (, ) σ s (, ) d d + ρ d s() σ d (, ) d σ s (, ) d, for any. In paricular, choosing =, +δ g s (, ) = σ s (, ) σ s (, ) d d, wo curves valuaion Quaniaive Produc Group, Sanander 9

12 Preliminary version Sepember 17, 29 and here are an infiniy of possible choices for he value of µ s (, ) wih [, + δ) saisfying for example +δ µ s (, ) d = g s (, ), µ s (, ) = σ s (, ) µ s (, ) = 1 δ +δ σ s (, a) σ s (, ) d, a or σ s (, ) d da, For + δ, we can differeniae in our condiion o ge ha µ s should saisfy wih g s (, ) = (σ s(, + δ) σ s (, )) +ρ d s()σ d (, ) µ s (, + δ) µ s (, ) = g s (, ), +δ For [ + iδ, + (i + 1)δ), we can wrie +δ σ s (, ) d σ s (, ) d + ρ d s() ec. σ d (, ) d (σ s (, + δ) σ s (, )). µ s (, ) = µ s (, δ) + g s (, ) = µ s(, 2δ) + g s (, 2δ) + g s (, δ) =... = i g s µ s (, iδ) + (, jδ). j=1 Wih his expression, we easily have ha condiion (3.7) is saisfied for any choice of he expression of µ s (, ) when is in [, + δ). 3.3 Foreign currency discoun curve i.f.r. drif For each foreign economy considered in our model, we assume he following diffusions for he domesic and foreign discouning i.f.r. and for he FX spo: (3.8) df d d (, ) := df d (, ) = µ d (, )d + σ d (, )dw d, df f d (, ) := df f (, ) = µ f (, )d + σ f (, )dw f, (3.9) ds() (3.1) S() = [f d d (, ) f f d (, )]d + σs ()dw S. Consider he quaniy S()B f (, ). Le us recall ha in his work, we are no discussing he consrucion of he discoun and esimaion curves. We consider ha foreign curves are consruced, for example, by using he mehodology proposed in [BS], and ha f f d (, ) is he spo curve used under P d measure o discoun all flows in he foreign economy. hus, B f (, ) = e f f (,s) ds wo curves valuaion Quaniaive Produc Group, Sanander 1

13 Preliminary version Sepember 17, 29 would be he price oday (in unis of foreign currency) of a conrac paying one uni of foreign currency a ime, seen from he domesic economy. ypically, his price would include he cross currency basis risk as shown in [BS]. herefore, naurally exending his noion o ime >, we see ha B f (, ) represens he price a ime of a conrac paying a uni of foreign currency a ime, expressed in foreign unis of currency, seen from he domesic economy. Muliplying i by S() ransforms i ino unis of domesic currency. he quaniy S()B f (, ) is herefore he price of a radable asse in he domesic economy. Applying Îo s lemma o he process and by using (3.1) and (2.4), we ge X = S()B f (, ), dx X = (f d (, ) f f (, )) d + (f f (, ) A f (, ) (Sf (, )) 2 ) d ρ f S ()σs ()S f (, ) d + σ S () dw S S f (, ) dw f, wih A f, S f as in (2.2). By no arbirage argumens, he drif in his Iô equaion mus be f d (, ), and herefore we impose (3.11) A f (, ) = 1 2 (Sf (, )) 2 ρ f S ()Sf (, )σ S (). Differeniaing in, his equaion gives (3.12) µ f (, ) = σ f (, ) σ f (, s)ds ρ f S σf (, )σ S (). 3.4 Foreign currency esimaion curve For each foreign economy considered in our model, we assume he following diffusions for he domesic and foreign discouning i.f.r., foreign reference i.f.r. and for he FX spo: (3.13) (3.14) (3.15) (3.16) df d d (, ) := µd d (, )d + σd dd d (, )dw, df f d (, ) := µf d (, )d + σf fd d (, )dw, df r f (, ) := µ f r (, )d + σr f (, )dw fr, ds() S() = [fd d (, ) f f d (, )]d + σs ()dw S. In he former secion, he condiion o be saisfied was (1.4). In he case of he foreign curves, we will require he same condiion, ha is, ( F RA f floaing (,, U) = EP e ) U r f d (s) ds L f r (,, U) = B f d (, U)Lf r (,, U), since he curves are consruced o saisfy his condiion a =. Observe ha his condiion is compleely analogous o he one imposed for he domesic reference curve wih respec o he domesic discoun curve. Following he same reasoning as in he domesic case (observe ha wo curves valuaion Quaniaive Produc Group, Sanander 11

14 Preliminary version Sepember 17, 29 no arbirage argumens were used), we ge ha he drifs of he reference curve (alernaively, spread curve) are as follows. For [ + iδ, + (i + 1)δ), i 1 wih µ f r (, ) = µ f r (, iδ) + g f r (, ) = (σf r (, + δ) σ f r (, )) +ρ fd fr ()σf d (, + δ) +δ +δ i j=1 gr f (, jδ), σ f r (, ) d d σ f r (, ) d + ρ fd fr () +δ σ f d (, ) d (σ f r (, + δ) σ f r (, )). he value of µ f r (, ) wih [, + δ) can be any of he number of posibiliies ha saisfy being +δ µ f r (, ) d = g f r (, ), +δ +δ +δ gr f (, ) = σr f (, ) σr f (, ) d d + ρ fd fr () σ f d (, ) d σr f (, ) d. In he case of he foreign spread curve, he same argumens give us ha for [ + iδ, + (i + 1)δ), i 1, wih µ f s (, ) = µ f s (, iδ) + g f s (, ) = (σf s (, + δ) σ f s (, )) +ρ fd fs ()σf d (, ) +δ +δ i j=1 σ f s (, ) d σ f s (, ) d + ρ fd fs () gs f (, jδ). σ f d (, ) d (σ f s (, + δ) σ f s (, )). he expression of µ f s (, ) wih [, + δ) is again any of he choices saisfying where +δ µ f s (, ) d = g f s (, ), +δ gs f (, ) = σs f (, ) σs f (, ) d d. 4 An example: dynamics in a Hull-Whie model he Hull-Whie model proposes he following sochasic differenial equaion for he shor rae: dr() = (θ() a()r()) d + σ()dw (). wo curves valuaion Quaniaive Produc Group, Sanander 12

15 Preliminary version Sepember 17, 29 We can solve he former equaion o ge: being We also define: Recall ha, in an HJM model, r() = µ() + b() f s dw s, b() = e asds, f() = σ() b(). β() = r() = f(, ) = f(, ) + b s ds. µ(s, )ds + and herefore, we can hen deduce ha, in he Hull-Whie conex: From his expression, (4.1) S(, ) = and S(u, ) dw (u) = β( ) σ(, ) = b()f(). σ(s, )dw (s), σ(, u) du = f()(β( ) β()) f(u) dw (u) f(u)β(u) dw (u). Observe ha his is valid for any curve wih a Hull-Whie dynamics. Also, he discussion above shows ha he requiremen given by equaion (1.4) is fulfilled wih a unique funcion µ α γ (, ) for each choice of γ = r, s, d and α = f, d. Once his is graned, from he poin of view of he valuaion wih his model, he quaniies whose dynamics need o be specified are bonds for he discoun curves and Libor raes for he esimaion and spread curves. 4.1 Domesic discouning bonds dynamics By condiion (3.1) and (4.1), he expressions for A d d is and herefore A d d (, ) = 1 2 (βd d ( ) βd d ())2 (f d d ())2, A d d (, ) d = 1 2 (βd d ( ))2 (fd d (s))2 ds βd d ( ) βd d (s)(f d d (s))2 ds+ 1 2 (β d d (s))2 (f d d (s))2 ds wo curves valuaion Quaniaive Produc Group, Sanander 13

16 Preliminary version Sepember 17, Domesic Libor dynamics In erms of he reference curve, we have o express he dynamics of ( ) B L d d r(;, U) = r (, ) Br d (, U) 1 1 m(, U), where we wrie Br d { (, ) Br d (, U) = Bd r (, ) Br d (, U) exp (A d r(u, U) A d r(u, )) du Condiion (3.5) and (4.1) impose A d r(u, U) A d r(u, ) herefore = 1 2 (Sd r (u, U) S d r (u, )) 2 + ρ dd dr (u)sd d (u, U)(Sd r (u, U) S d r (u, )) } (Sr d (u, U) Sr d (u, )) dw dr (u). = 1 2 (f d r (u)) 2 (β d r (U) β d r ( )) 2 + ρ dd dr (u)f d r (u)f d d (u)(βd d (U) βd d (u))(βd r (U) β d r ( )) (A d r(u, U) A d r(u, )) du = 1 2 (βd r (U) βr d ( )) 2 (fr d (u)) 2 du +(βr d (U) βr d ( )) (β dd (U) ρ dd dr (u)f r d (u)fd d (u) du In erms of he spread curve, we have o express he dynamics of ( B L d d r(;, U) = d (, )Bs d ) (, ) Bd d(, U)Bd s (, U) 1 1 m(, U), where we wrie Bs d { (, ) Bs d (, U) = Bd s (, ) Bs d (, U) exp (A d s(u, U) A d s(u, )) du Condiion (3.7) and (4.1), impose A d s(u, U) A d s(u, ) herefore = 1 2 (Sd s (u, U) S d s (u, )) 2 + ρ dd ds (u)sd d (u, )(Sd s (u, U) S d s (u, )) ρ dd dr (u)f d r (u)f d d (u)βd d (u) du ) } (Ss d (u, U) Ss d (u, )) dw ds (u). = 1 2 (f d s (u)) 2 (β d s (U) β d s ( )) 2 + ρ dd ds (u)f d s (u)f d d (u)(βd d ( ) βd d (u))(βd s (U) β d s ( )) (A d s(u, U) A d s(u, )) du = 1 2 (βd s (U) βs d ( )) 2 (fs d (u)) 2 du +(βs d (U) βs d ( )) (β dd ( ) ρ dd dr (u)f s d (u)fd d (u) du ρ dd dr (u)f d s (u)f d d (u)βd d (u) du ). wo curves valuaion Quaniaive Produc Group, Sanander 14

17 Preliminary version Sepember 17, Foreign discouning bond dynamics By condiion (3.11) and (4.1), he expression for A f d is herefore A f d (, ) = 1 2 (f f d ())2 (β f d ( ) βf d ())2 ρ fd S ()σ S()f f d ()(βf d ( ) βf d ()). A f d (u, ) du = 1 2 (βf d ( ))2 (f f d (u))2 du β f d ( ) (f f d (u))2 β f d (u) du β f d ( ) ρ fd S (u)σ S(u)f f d (u) du Foreign Libor dynamics ρ fd S (u)σ S(u)f f d (u)βf d (u) du. (f f d (u))2 (β f d (u))2 du Analogously o he domesic Libor dynamics, in erms of he reference curve, we have o express he dynamics of L f r (;, U) = ( B f ) r (, ) Br f (, U) 1 1 m(, U), where we wrie Br f (, ) Br f (, U) = B f { r (, ) Br f (, U) exp (A f r (u, U) A f r (u, )) du Condiion (3.5) imposes A f r (u, U) A f r (u, ) = 1 2 (Sf r (u, U) S f r (u, )) 2 + ρ fd fr (u)sf d (u, U)(Sf r (u, U) S f r (u, )) } (Sr f (u, U) Sr f (u, )) dw fr (u). = 1 2 (f f r (u)) 2 (β f r (U) β f r ( )) 2 + ρ fd fr (u)f f r (u)f f d (u)(βf d (U) βf d (u))(βf r (U) β f r ( )) herefore (A f r (u, U) A f r (u, )) du = 1 2 (βf r (U) βr f ( )) 2 (fr f (u)) 2 du +(βr f (U) βr f ( )) (β fd (U) ρ fd fr (u)f r f (u)f f d (u) du In erms of he spread curve, we have o express he dynamics of L f r (;, U) = ( B f d (, ) )Bf s (, ) B f d (, U)Bf s (, U) 1 1 m(, U), ρ fd fr (u)f f r (u)f f d (u)βf d (u) du ) wo curves valuaion Quaniaive Produc Group, Sanander 15

18 Preliminary version Sepember 17, 29 where we wrie Bs f (, ) Bs f (, U) = B f { s (, ) Bs f (, U) exp (A f s (u, U) A f s (u, )) du Condiion (3.7) imposes A f s (u, U) A f s (u, ) = 1 2 (Sf s (u, U) S f s (u, )) 2 + ρ fd fs (u)sf d (u, )(Sf s (u, U) S f s (u, )) } (Ss f (u, U) Ss f (u, )) dw ds (u). = 1 2 (f f s (u)) 2 (β f s (U) β f s ( )) 2 + ρ fd fs (u)f f s (u)f f d (u)(βf d ( ) βf d (u))(βf s (U) β f s ( )) herefore (A f s (u, U) A f s (u, )) du = 1 2 (βf s (U) βs f ( )) 2 (fs f (u)) 2 du +(βs f (U) βs f ( )) (β fd ( ) ρ fd fr (u)f s f (u)f f d (u) du References ρ fd fr (u)f f s (u)f f d (u)βf d (u) du ). [AS] Acerbi, C., Scandolo, G., Liquidiy risk heory and coheren measures of risk, Quaniaive Finance, 8 (28) [Bi] Bianchei, M. wo curves, one price: pricing and hedging ineres rae derivaives using differen yield curves for discouning and forwarding, preprin, available on line a hp://www1.mae.polimi.i/wqf9/wqf9/quaniaive_finance/ 29/Aula_D/4-Fixed_Income/2-A-Bianchei-DoubleCurvePricing-v1.4.pdf [Bj] Björk,. Arbirage heory in Coninuous ime, Oxford Finance, Oxford Universiy Press, 24. [BS] Boenkos, W., Schmid, W., Cross currency swap valuaion, preprin, available on line a hp:// Arbeis2.pdf [BM] [ChFM] Brigo, D., Mercurio, F. Ineres rae models. heory and pracice, Springer Finance, Berlin, 21. Chiarella, C, Fanelli, V., Musi, S., Modelling evoluion of Credi Spreads using he Cox process wihin he HJM framework: a CDS opion pricing model. Research Paper Series 232, Quaniaive Finance Research Cenre, Universiy of echnology, Sydney. Available on line a hp:// research_papers/rp232.pdf. wo curves valuaion Quaniaive Produc Group, Sanander 16

19 Preliminary version Sepember 17, 29 [C] [DS] [JP] [KS] [KW] [M] [S1] [S2] [] Chow, Y.S., eicher, H. Probabiliy heory. Independence, Inerchangeabiliy, Maringales, Springer-Verlag, New York, Duffie, D., Singleon, K.J., Modeling erm Srucures of Defaulable Bonds, he Review of Financial Sudies, 12 (1999). Jarrow, R.A., Proer, P., Liquidiy Risk and Opion Pricing heory, in Handbook in Operaions Research and Managemen Science, Financial Engineering 15, , J. Birge and V. Linesky, eds., Norh Holland, 27. Karazas, I., Shreve, S.,Brownian moion and sochasic calculus, Graduae exs in Mahs., 113, Springer-Verlag, Berlin, Kijima, M., anaka, M., Wong,., A muli-qualiy model of ineres raes, Quaniaive Finance 9, Mercurio, F. Ineres Raes and he Credi Crunch: New Formulas and Marke Models. Preprin. Bloomberg, QFR. Available on line a hp:// i/lmmposcrunch5.pdf Schönbucker, P.J., A Libor Marke Model wih defaul risk, Bonn Econ Discussion Papers,15/21, Bonn Graduae School of Economics, Deparmen of Economics, Universiy of Bonn. Available on line a fp://web.bgse.uni-bonn.de/pub/repec/ bon/bonedp/bgse15_21.pdf. Schönbucker, P.J., Credi Derivaives pricing models. Models, Pricing and Implemenaion, Wiley Finance Series, John Wiley and Sons, England, 23. anaka, K., Heerogeneous Yield Curves and Basis Swaps. Preprin. Available on line a hp://www2.econ.osaka-u.ac.jp/library/global/dp/312.pdf wo curves valuaion Quaniaive Produc Group, Sanander 17